null hypothesis p value example

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StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2024 Jan-.

StatPearls [Internet].

Hypothesis testing, p values, confidence intervals, and significance.

Jacob Shreffler ; Martin R. Huecker .

Affiliations

Last Update: March 13, 2023 .

Definition/Introduction

Medical providers often rely on evidence-based medicine to guide decision-making in practice. Often a research hypothesis is tested with results provided, typically with p values, confidence intervals, or both. Additionally, statistical or research significance is estimated or determined by the investigators. Unfortunately, healthcare providers may have different comfort levels in interpreting these findings, which may affect the adequate application of the data.

Issues of Concern

Without a foundational understanding of hypothesis testing, p values, confidence intervals, and the difference between statistical and clinical significance, it may affect healthcare providers' ability to make clinical decisions without relying purely on the research investigators deemed level of significance. Therefore, an overview of these concepts is provided to allow medical professionals to use their expertise to determine if results are reported sufficiently and if the study outcomes are clinically appropriate to be applied in healthcare practice.

Hypothesis Testing

Investigators conducting studies need research questions and hypotheses to guide analyses. Starting with broad research questions (RQs), investigators then identify a gap in current clinical practice or research. Any research problem or statement is grounded in a better understanding of relationships between two or more variables. For this article, we will use the following research question example:

Research Question: Is Drug 23 an effective treatment for Disease A?

Research questions do not directly imply specific guesses or predictions; we must formulate research hypotheses. A hypothesis is a predetermined declaration regarding the research question in which the investigator(s) makes a precise, educated guess about a study outcome. This is sometimes called the alternative hypothesis and ultimately allows the researcher to take a stance based on experience or insight from medical literature. An example of a hypothesis is below.

Research Hypothesis: Drug 23 will significantly reduce symptoms associated with Disease A compared to Drug 22.

The null hypothesis states that there is no statistical difference between groups based on the stated research hypothesis.

Researchers should be aware of journal recommendations when considering how to report p values, and manuscripts should remain internally consistent.

Regarding p values, as the number of individuals enrolled in a study (the sample size) increases, the likelihood of finding a statistically significant effect increases. With very large sample sizes, the p-value can be very low significant differences in the reduction of symptoms for Disease A between Drug 23 and Drug 22. The null hypothesis is deemed true until a study presents significant data to support rejecting the null hypothesis. Based on the results, the investigators will either reject the null hypothesis (if they found significant differences or associations) or fail to reject the null hypothesis (they could not provide proof that there were significant differences or associations).

To test a hypothesis, researchers obtain data on a representative sample to determine whether to reject or fail to reject a null hypothesis. In most research studies, it is not feasible to obtain data for an entire population. Using a sampling procedure allows for statistical inference, though this involves a certain possibility of error. [1] When determining whether to reject or fail to reject the null hypothesis, mistakes can be made: Type I and Type II errors. Though it is impossible to ensure that these errors have not occurred, researchers should limit the possibilities of these faults. [2]

Significance

Significance is a term to describe the substantive importance of medical research. Statistical significance is the likelihood of results due to chance. [3] Healthcare providers should always delineate statistical significance from clinical significance, a common error when reviewing biomedical research. [4] When conceptualizing findings reported as either significant or not significant, healthcare providers should not simply accept researchers' results or conclusions without considering the clinical significance. Healthcare professionals should consider the clinical importance of findings and understand both p values and confidence intervals so they do not have to rely on the researchers to determine the level of significance. [5] One criterion often used to determine statistical significance is the utilization of p values.

P values are used in research to determine whether the sample estimate is significantly different from a hypothesized value. The p-value is the probability that the observed effect within the study would have occurred by chance if, in reality, there was no true effect. Conventionally, data yielding a p<0.05 or p<0.01 is considered statistically significant. While some have debated that the 0.05 level should be lowered, it is still universally practiced. [6] Hypothesis testing allows us to determine the size of the effect.

An example of findings reported with p values are below:

Statement: Drug 23 reduced patients' symptoms compared to Drug 22. Patients who received Drug 23 (n=100) were 2.1 times less likely than patients who received Drug 22 (n = 100) to experience symptoms of Disease A, p<0.05.

Statement:Individuals who were prescribed Drug 23 experienced fewer symptoms (M = 1.3, SD = 0.7) compared to individuals who were prescribed Drug 22 (M = 5.3, SD = 1.9). This finding was statistically significant, p= 0.02.

For either statement, if the threshold had been set at 0.05, the null hypothesis (that there was no relationship) should be rejected, and we should conclude significant differences. Noticeably, as can be seen in the two statements above, some researchers will report findings with < or > and others will provide an exact p-value (0.000001) but never zero [6] . When examining research, readers should understand how p values are reported. The best practice is to report all p values for all variables within a study design, rather than only providing p values for variables with significant findings. [7] The inclusion of all p values provides evidence for study validity and limits suspicion for selective reporting/data mining.

While researchers have historically used p values, experts who find p values problematic encourage the use of confidence intervals. [8] . P-values alone do not allow us to understand the size or the extent of the differences or associations. [3] In March 2016, the American Statistical Association (ASA) released a statement on p values, noting that scientific decision-making and conclusions should not be based on a fixed p-value threshold (e.g., 0.05). They recommend focusing on the significance of results in the context of study design, quality of measurements, and validity of data. Ultimately, the ASA statement noted that in isolation, a p-value does not provide strong evidence. [9]

When conceptualizing clinical work, healthcare professionals should consider p values with a concurrent appraisal study design validity. For example, a p-value from a double-blinded randomized clinical trial (designed to minimize bias) should be weighted higher than one from a retrospective observational study [7] . The p-value debate has smoldered since the 1950s [10] , and replacement with confidence intervals has been suggested since the 1980s. [11]

Confidence Intervals

A confidence interval provides a range of values within given confidence (e.g., 95%), including the accurate value of the statistical constraint within a targeted population. [12] Most research uses a 95% CI, but investigators can set any level (e.g., 90% CI, 99% CI). [13] A CI provides a range with the lower bound and upper bound limits of a difference or association that would be plausible for a population. [14] Therefore, a CI of 95% indicates that if a study were to be carried out 100 times, the range would contain the true value in 95, [15] confidence intervals provide more evidence regarding the precision of an estimate compared to p-values. [6]

In consideration of the similar research example provided above, one could make the following statement with 95% CI:

Statement: Individuals who were prescribed Drug 23 had no symptoms after three days, which was significantly faster than those prescribed Drug 22; there was a mean difference between the two groups of days to the recovery of 4.2 days (95% CI: 1.9 – 7.8).

It is important to note that the width of the CI is affected by the standard error and the sample size; reducing a study sample number will result in less precision of the CI (increase the width). [14] A larger width indicates a smaller sample size or a larger variability. [16] A researcher would want to increase the precision of the CI. For example, a 95% CI of 1.43 – 1.47 is much more precise than the one provided in the example above. In research and clinical practice, CIs provide valuable information on whether the interval includes or excludes any clinically significant values. [14]

Null values are sometimes used for differences with CI (zero for differential comparisons and 1 for ratios). However, CIs provide more information than that. [15] Consider this example: A hospital implements a new protocol that reduced wait time for patients in the emergency department by an average of 25 minutes (95% CI: -2.5 – 41 minutes). Because the range crosses zero, implementing this protocol in different populations could result in longer wait times; however, the range is much higher on the positive side. Thus, while the p-value used to detect statistical significance for this may result in "not significant" findings, individuals should examine this range, consider the study design, and weigh whether or not it is still worth piloting in their workplace.

Similarly to p-values, 95% CIs cannot control for researchers' errors (e.g., study bias or improper data analysis). [14] In consideration of whether to report p-values or CIs, researchers should examine journal preferences. When in doubt, reporting both may be beneficial. [13] An example is below:

Reporting both: Individuals who were prescribed Drug 23 had no symptoms after three days, which was significantly faster than those prescribed Drug 22, p = 0.009. There was a mean difference between the two groups of days to the recovery of 4.2 days (95% CI: 1.9 – 7.8).

Clinical Significance

Recall that clinical significance and statistical significance are two different concepts. Healthcare providers should remember that a study with statistically significant differences and large sample size may be of no interest to clinicians, whereas a study with smaller sample size and statistically non-significant results could impact clinical practice. [14] Additionally, as previously mentioned, a non-significant finding may reflect the study design itself rather than relationships between variables.

Healthcare providers using evidence-based medicine to inform practice should use clinical judgment to determine the practical importance of studies through careful evaluation of the design, sample size, power, likelihood of type I and type II errors, data analysis, and reporting of statistical findings (p values, 95% CI or both). [4] Interestingly, some experts have called for "statistically significant" or "not significant" to be excluded from work as statistical significance never has and will never be equivalent to clinical significance. [17]

The decision on what is clinically significant can be challenging, depending on the providers' experience and especially the severity of the disease. Providers should use their knowledge and experiences to determine the meaningfulness of study results and make inferences based not only on significant or insignificant results by researchers but through their understanding of study limitations and practical implications.

Nursing, Allied Health, and Interprofessional Team Interventions

All physicians, nurses, pharmacists, and other healthcare professionals should strive to understand the concepts in this chapter. These individuals should maintain the ability to review and incorporate new literature for evidence-based and safe care.

Review Questions
Access free multiple choice questions on this topic.
Comment on this article.

Disclosure: Jacob Shreffler declares no relevant financial relationships with ineligible companies.

Disclosure: Martin Huecker declares no relevant financial relationships with ineligible companies.

This book is distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ), which permits others to distribute the work, provided that the article is not altered or used commercially. You are not required to obtain permission to distribute this article, provided that you credit the author and journal.

Cite this Page Shreffler J, Huecker MR. Hypothesis Testing, P Values, Confidence Intervals, and Significance. [Updated 2023 Mar 13]. In: StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2024 Jan-.

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S.3.2 hypothesis testing (p-value approach).

The P -value approach involves determining "likely" or "unlikely" by determining the probability — assuming the null hypothesis was true — of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed. If the P -value is small, say less than (or equal to) $\alpha$, then it is "unlikely." And, if the P -value is large, say more than $\alpha$, then it is "likely."

If the P -value is less than (or equal to) $\alpha$, then the null hypothesis is rejected in favor of the alternative hypothesis. And, if the P -value is greater than $\alpha$, then the null hypothesis is not rejected.

Specifically, the four steps involved in using the P -value approach to conducting any hypothesis test are:

Specify the null and alternative hypotheses.
Using the sample data and assuming the null hypothesis is true, calculate the value of the test statistic. Again, to conduct the hypothesis test for the population mean μ , we use the t -statistic $t^*=\frac{\bar{x}-\mu}{s/\sqrt{n}}$ which follows a t -distribution with n - 1 degrees of freedom.
Using the known distribution of the test statistic, calculate the P -value : "If the null hypothesis is true, what is the probability that we'd observe a more extreme test statistic in the direction of the alternative hypothesis than we did?" (Note how this question is equivalent to the question answered in criminal trials: "If the defendant is innocent, what is the chance that we'd observe such extreme criminal evidence?")
Set the significance level, $\alpha$, the probability of making a Type I error to be small — 0.01, 0.05, or 0.10. Compare the P -value to $\alpha$. If the P -value is less than (or equal to) $\alpha$, reject the null hypothesis in favor of the alternative hypothesis. If the P -value is greater than $\alpha$, do not reject the null hypothesis.

Example S.3.2.1

Mean gpa section .

In our example concerning the mean grade point average, suppose that our random sample of n = 15 students majoring in mathematics yields a test statistic t * equaling 2.5. Since n = 15, our test statistic t * has n - 1 = 14 degrees of freedom. Also, suppose we set our significance level α at 0.05 so that we have only a 5% chance of making a Type I error.

Right Tailed

The P -value for conducting the right-tailed test H 0 : μ = 3 versus H A : μ > 3 is the probability that we would observe a test statistic greater than t * = 2.5 if the population mean $\mu$ really were 3. Recall that probability equals the area under the probability curve. The P -value is therefore the area under a t n - 1 = t 14 curve and to the right of the test statistic t * = 2.5. It can be shown using statistical software that the P -value is 0.0127. The graph depicts this visually.

t-distrbution graph showing the right tail beyond a t value of 2.5

The P -value, 0.0127, tells us it is "unlikely" that we would observe such an extreme test statistic t * in the direction of H A if the null hypothesis were true. Therefore, our initial assumption that the null hypothesis is true must be incorrect. That is, since the P -value, 0.0127, is less than $\alpha$ = 0.05, we reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ > 3.

Note that we would not reject H 0 : μ = 3 in favor of H A : μ > 3 if we lowered our willingness to make a Type I error to $\alpha$ = 0.01 instead, as the P -value, 0.0127, is then greater than $\alpha$ = 0.01.

Left Tailed

In our example concerning the mean grade point average, suppose that our random sample of n = 15 students majoring in mathematics yields a test statistic t * instead of equaling -2.5. The P -value for conducting the left-tailed test H 0 : μ = 3 versus H A : μ < 3 is the probability that we would observe a test statistic less than t * = -2.5 if the population mean μ really were 3. The P -value is therefore the area under a t n - 1 = t 14 curve and to the left of the test statistic t* = -2.5. It can be shown using statistical software that the P -value is 0.0127. The graph depicts this visually.

t distribution graph showing left tail below t value of -2.5

The P -value, 0.0127, tells us it is "unlikely" that we would observe such an extreme test statistic t * in the direction of H A if the null hypothesis were true. Therefore, our initial assumption that the null hypothesis is true must be incorrect. That is, since the P -value, 0.0127, is less than α = 0.05, we reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ < 3.

Note that we would not reject H 0 : μ = 3 in favor of H A : μ < 3 if we lowered our willingness to make a Type I error to α = 0.01 instead, as the P -value, 0.0127, is then greater than $\alpha$ = 0.01.

In our example concerning the mean grade point average, suppose again that our random sample of n = 15 students majoring in mathematics yields a test statistic t * instead of equaling -2.5. The P -value for conducting the two-tailed test H 0 : μ = 3 versus H A : μ ≠ 3 is the probability that we would observe a test statistic less than -2.5 or greater than 2.5 if the population mean μ really was 3. That is, the two-tailed test requires taking into account the possibility that the test statistic could fall into either tail (hence the name "two-tailed" test). The P -value is, therefore, the area under a t n - 1 = t 14 curve to the left of -2.5 and to the right of 2.5. It can be shown using statistical software that the P -value is 0.0127 + 0.0127, or 0.0254. The graph depicts this visually.

t-distribution graph of two tailed probability for t values of -2.5 and 2.5

Note that the P -value for a two-tailed test is always two times the P -value for either of the one-tailed tests. The P -value, 0.0254, tells us it is "unlikely" that we would observe such an extreme test statistic t * in the direction of H A if the null hypothesis were true. Therefore, our initial assumption that the null hypothesis is true must be incorrect. That is, since the P -value, 0.0254, is less than α = 0.05, we reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ ≠ 3.

Note that we would not reject H 0 : μ = 3 in favor of H A : μ ≠ 3 if we lowered our willingness to make a Type I error to α = 0.01 instead, as the P -value, 0.0254, is then greater than $\alpha$ = 0.01.

Now that we have reviewed the critical value and P -value approach procedures for each of the three possible hypotheses, let's look at three new examples — one of a right-tailed test, one of a left-tailed test, and one of a two-tailed test.

The good news is that, whenever possible, we will take advantage of the test statistics and P -values reported in statistical software, such as Minitab, to conduct our hypothesis tests in this course.

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How to Find the P value: Process and Calculations

By Jim Frost 4 Comments

P values are everywhere in statistics . They’re in all types of hypothesis tests. But how do you calculate a p-value ? Unsurprisingly, the precise calculations depend on the test. However, there is a general process that applies to finding a p value.

In this post, you’ll learn how to find the p value. I’ll start by showing you the general process for all hypothesis tests. Then I’ll move on to a step-by-step example showing the calculations for a p value. This post includes a calculator so you can apply what you learn.

General Process for How to Find the P value

To find the p value for your sample , do the following:

Identify the correct test statistic.
Calculate the test statistic using the relevant properties of your sample.
Specify the characteristics of the test statistic’s sampling distribution.
Place your test statistic in the sampling distribution to find the p value.

Before moving on to the calculations example, I’ll summarize the purpose for each step. This part tells you the “why.” In the example calculations section, I show the “how.”

Identify the Correct Test Statistic

All hypothesis tests boil your sample data down to a single number known as a test statistic. T-tests use t-values. F-tests use F-values. Chi-square tests use chi-square values. Choosing the correct one depends on the type of data you have and how you want to analyze it. Before you can find the p value, you must determine which hypothesis test and test statistic you’ll use.

Test statistics assess how consistent your sample data are with the null hypothesis. As a test statistic becomes more extreme, it indicates a larger difference between your sample data and the null hypothesis.

Calculate the Test Statistic

How you calculate the test statistic depends on which one you’re using. Unsurprisingly, the method for calculating test statistics varies by test type. Consequently, to calculate the p value for any test, you’ll need to know the correct test statistic formula.

To learn more about test statistics and how to calculate them for other tests, read my article, Test Statistics .

Specify the Properties of the Test Statistic’s Sampling Distribution

Test statistics are unitless, making them tricky to interpret on their own. You need to place them in a larger context to understand how extreme they are.

The sampling distribution for the test statistic provides that context. Sampling distributions are a type of probability distribution. Consequently, they allow you to calculate probabilities related to your test statistic’s extremeness, which lets us find the p value!

Probability distribution plot that displays a t-distribution.

Like any distribution, the same sampling distribution (e.g., the t-distribution) can have a variety of shapes depending upon its parameters . For this step, you need to determine the characteristics of the sampling distribution that fit your design and data.

That usually entails specifying the degrees of freedom (changes its shape) and whether the test is one- or two-tailed (affects the directions the test can detect effects). In essence, you’re taking the general sampling distribution and tailoring it to your study so it provides the correct probabilities for finding the p value.

Each test statistic’s sampling distribution has unique properties you need to specify. At the end of this post, I provide links for several.

Learn more about degrees of freedom and one-tailed vs. two-tailed tests .

Placing Your Test Statistic in its Sampling Distribution to Find the P value

Finally, it’s time to find the p value because we have everything in place. We have calculated our test statistic and determined the correct properties for its sampling distribution. Now, we need to find the probability of values more extreme than our observed test statistic.

In this context, more extreme means further away from the null value in both directions for a two-tailed test or in one direction for a one-tailed test.

At this point, there are two ways to use the test statistic and distribution to calculate the p value. The formulas for probability distributions are relatively complex. Consequently, you won’t calculate it directly. Instead, you’ll use either an online calculator or a statistical table for the test statistic. I’ll show you both approaches in the step-by-step example.

In summary, calculating a p-value involves identifying and calculating your test statistic and then placing it in its sampling distribution to find the probability of more extreme values!

Let’s see this whole process in action with an example!

Step-by-Step Example of How to Find the P value for a T-test

For this example, assume we’re tasked with determining whether a sample mean is different from a hypothesized value. We’re given the sample statistics below and need to find the p value.

Mean: 330.6
Standard deviation: 154.2
Sample size: 25
Null hypothesis value: 260

Let’s work through the step-by-step process of how to calculate a p-value.

First, we need to identify the correct test statistic. Because we’re comparing one mean to a null value, we need to use a 1-sample t-test. Hence, the t-value is our test statistic, and the t-distribution is our sampling distribution.

Second, we’ll calculate the test statistic. The t-value formula for a 1-sample t-test is the following:

x̄ is the sample mean.
µ 0 is the null hypothesis value.
s is the sample standard deviation.
n is the sample size
Collectively, the denominator is the standard error of the mean .

Let’s input our sample values into the equation to calculate the t-value.

Third, we need to specify the properties of the sampling distribution to find the p value. We’ll need the degrees of freedom.

The degrees of freedom for a 1-sample t-test is n – 1. Our sample size is 25. Hence, we have 24 DF. We’ll use a two-tailed test, which is the standard.

Now we’ve got all the necessary information to calculate the p-value. I’ll show you two ways to take the final step!

P-value Calculator

One method is to use an online p-value calculator, like the one I include below.

Enter the following in the calculator for our t-test example.

In What do you want? , choose Two-tailed p-value (the default).
In What do you have? , choose t-score .
In Degrees of freedom (d) , enter 24 .
In Your t-score , enter 2.289 .

The calculator displays a result of 0.031178.

There you go! Using the standard significance level of 0.05, our results are statistically significant!

Using a Statistical Table to Find the P Value

The other common method is using a statistical table. In this case, we’ll need to use a t-table. For this example, I’ll truncate the rows. You can find my full table here: T-Table .

This method won’t find the exact p value, but you’ll find a range and know whether your results are statistically significant.

Start by looking in the row for 24 degrees of freedom, highlighted in light green. We need to find where our t-score of 2.289 fits in. I highlight the two table values that our t-value fits between, 2.064 and 2.492. Then we look at the two-tailed row at the top to find the corresponding p values for the two t-values.

In this case, our t-value of 2.289 produces a p value between 0.02 and 0.05 for a two-tailed test. Our results are statistically significant, and they are consistent with the calculator’s more precise results.

Displaying the P value in a Chart

In the example above, you saw how to calculate a p-value starting with the sample statistics. We calculated the t-value and placed it in the applicable t-distribution. I find that the calculations and numbers are dry by themselves. I love graphing things whenever possible, so I’ll use a probability distribution plot to illustrate the example.

Using statistical software, I’ll create the graphical equivalent of calculating the p-value above.

This chart has two shaded regions because we performed a two-tailed test. Each region has a probability of 0.01559. When you sum them, you obtain the p-value of 0.03118. In other words, the likelihood of a t-value falling in either shaded region when the null hypothesis is true is 0.03118.

I showed you how to find the p value for a t-test. Click the links below to see how it works for other hypothesis tests:

One-Way ANOVA F-test
Chi-square Test of Independence

Now that we’ve found the p value, how do you interpret it precisely? If you’re going beyond the significant/not significant decision and really want to understand what it means, read my posts, Interpreting P Values and Statistical Significance: Definition & Meaning .

If you’re learning about hypothesis testing and like the approach I use in my blog, check out my Hypothesis Testing book! You can find it at Amazon and other retailers.

Cover image of my Hypothesis Testing: An Intuitive Guide ebook.

Reader Interactions

January 9, 2024 at 9:58 am

how did you get the 0.01559? is it from the t table or somewhere else. please put me through

January 9, 2024 at 3:13 pm

The value of 0.01559 comes from the t-distribution. It’s the probability of each red shaded region in the graph I show. These regions are based on the t-value. Typically, you’ll use either statistical software or a t-distribution calculator to find probabilities associated with t-values. Or use a t-table. I used my statistical software. You don’t calculate those probabilities yourself because the calculations are complex.

I hope that helps!

November 23, 2022 at 2:08 am

Simply superb. Easy for us who are starters to enjoy statistic made enjoyable.

November 22, 2022 at 6:41 pm

I like the way your presentation so that every one can undersanf in the simplest way. If you can support this by power point it will be more intetrsted. I know it takes your valuable time. However, forwarding your knowledge to those who need is more valuable, supporting and appreciation. Continue doing this teaching approach. Thank you. I wish you all the best. God bless you.

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P-Value in Statistical Hypothesis Tests: What is it?

P value definition.

A p value is used in hypothesis testing to help you support or reject the null hypothesis . The p value is the evidence against a null hypothesis . The smaller the p-value, the stronger the evidence that you should reject the null hypothesis.

P values are expressed as decimals although it may be easier to understand what they are if you convert them to a percentage . For example, a p value of 0.0254 is 2.54%. This means there is a 2.54% chance your results could be random (i.e. happened by chance). That’s pretty tiny. On the other hand, a large p-value of .9(90%) means your results have a 90% probability of being completely random and not due to anything in your experiment. Therefore, the smaller the p-value, the more important (“ significant “) your results.

When you run a hypothesis test , you compare the p value from your test to the alpha level you selected when you ran the test. Alpha levels can also be written as percentages.

P Value vs Alpha level

Alpha levels are controlled by the researcher and are related to confidence levels . You get an alpha level by subtracting your confidence level from 100%. For example, if you want to be 98 percent confident in your research, the alpha level would be 2% (100% – 98%). When you run the hypothesis test, the test will give you a value for p. Compare that value to your chosen alpha level. For example, let’s say you chose an alpha level of 5% (0.05). If the results from the test give you:

A small p (≤ 0.05), reject the null hypothesis . This is strong evidence that the null hypothesis is invalid.
A large p (> 0.05) means the alternate hypothesis is weak, so you do not reject the null.

P Values and Critical Values

What if I Don’t Have an Alpha Level?

In an ideal world, you’ll have an alpha level. But if you do not, you can still use the following rough guidelines in deciding whether to support or reject the null hypothesis:

If p > .10 → “not significant”
If p ≤ .10 → “marginally significant”
If p ≤ .05 → “significant”
If p ≤ .01 → “highly significant.”

How to Calculate a P Value on the TI 83

Example question: The average wait time to see an E.R. doctor is said to be 150 minutes. You think the wait time is actually less. You take a random sample of 30 people and find their average wait is 148 minutes with a standard deviation of 5 minutes. Assume the distribution is normal. Find the p value for this test.

Press STAT then arrow over to TESTS.
Press ENTER for Z-Test .
Arrow over to Stats. Press ENTER.
Arrow down to μ0 and type 150. This is our null hypothesis mean.
Arrow down to σ. Type in your std dev: 5.
Arrow down to xbar. Type in your sample mean : 148.
Arrow down to n. Type in your sample size : 30.
Arrow to <μ0 for a left tail test . Press ENTER.
Arrow down to Calculate. Press ENTER. P is given as .014, or about 1%.

The probability that you would get a sample mean of 148 minutes is tiny, so you should reject the null hypothesis.

Note : If you don’t want to run a test, you could also use the TI 83 NormCDF function to get the area (which is the same thing as the probability value).

Dodge, Y. (2008). The Concise Encyclopedia of Statistics . Springer. Gonick, L. (1993). The Cartoon Guide to Statistics . HarperPerennial.

What is The Null Hypothesis & When Do You Reject The Null Hypothesis

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A null hypothesis is a statistical concept suggesting no significant difference or relationship between measured variables. It’s the default assumption unless empirical evidence proves otherwise.

The null hypothesis states no relationship exists between the two variables being studied (i.e., one variable does not affect the other).

The null hypothesis is the statement that a researcher or an investigator wants to disprove.

Testing the null hypothesis can tell you whether your results are due to the effects of manipulating the dependent variable or due to random chance.

How to Write a Null Hypothesis

Null hypotheses (H0) start as research questions that the investigator rephrases as statements indicating no effect or relationship between the independent and dependent variables.

It is a default position that your research aims to challenge or confirm.

For example, if studying the impact of exercise on weight loss, your null hypothesis might be:

There is no significant difference in weight loss between individuals who exercise daily and those who do not.

Examples of Null Hypotheses

Research Question	Null Hypothesis
Do teenagers use cell phones more than adults?	Teenagers and adults use cell phones the same amount.
Do tomato plants exhibit a higher rate of growth when planted in compost rather than in soil?	Tomato plants show no difference in growth rates when planted in compost rather than soil.
Does daily meditation decrease the incidence of depression?	Daily meditation does not decrease the incidence of depression.
Does daily exercise increase test performance?	There is no relationship between daily exercise time and test performance.
Does the new vaccine prevent infections?	The vaccine does not affect the infection rate.
Does flossing your teeth affect the number of cavities?	Flossing your teeth has no effect on the number of cavities.

When Do We Reject The Null Hypothesis?

We reject the null hypothesis when the data provide strong enough evidence to conclude that it is likely incorrect. This often occurs when the p-value (probability of observing the data given the null hypothesis is true) is below a predetermined significance level.

If the collected data does not meet the expectation of the null hypothesis, a researcher can conclude that the data lacks sufficient evidence to back up the null hypothesis, and thus the null hypothesis is rejected.

Rejecting the null hypothesis means that a relationship does exist between a set of variables and the effect is statistically significant ( p > 0.05).

If the data collected from the random sample is not statistically significance , then the null hypothesis will be accepted, and the researchers can conclude that there is no relationship between the variables.

You need to perform a statistical test on your data in order to evaluate how consistent it is with the null hypothesis. A p-value is one statistical measurement used to validate a hypothesis against observed data.

Calculating the p-value is a critical part of null-hypothesis significance testing because it quantifies how strongly the sample data contradicts the null hypothesis.

The level of statistical significance is often expressed as a p -value between 0 and 1. The smaller the p-value, the stronger the evidence that you should reject the null hypothesis.

Probability and statistical significance in ab testing. Statistical significance in a b experiments

Usually, a researcher uses a confidence level of 95% or 99% (p-value of 0.05 or 0.01) as general guidelines to decide if you should reject or keep the null.

When your p-value is less than or equal to your significance level, you reject the null hypothesis.

In other words, smaller p-values are taken as stronger evidence against the null hypothesis. Conversely, when the p-value is greater than your significance level, you fail to reject the null hypothesis.

In this case, the sample data provides insufficient data to conclude that the effect exists in the population.

Because you can never know with complete certainty whether there is an effect in the population, your inferences about a population will sometimes be incorrect.

When you incorrectly reject the null hypothesis, it’s called a type I error. When you incorrectly fail to reject it, it’s called a type II error.

Why Do We Never Accept The Null Hypothesis?

The reason we do not say “accept the null” is because we are always assuming the null hypothesis is true and then conducting a study to see if there is evidence against it. And, even if we don’t find evidence against it, a null hypothesis is not accepted.

A lack of evidence only means that you haven’t proven that something exists. It does not prove that something doesn’t exist.

It is risky to conclude that the null hypothesis is true merely because we did not find evidence to reject it. It is always possible that researchers elsewhere have disproved the null hypothesis, so we cannot accept it as true, but instead, we state that we failed to reject the null.

One can either reject the null hypothesis, or fail to reject it, but can never accept it.

Why Do We Use The Null Hypothesis?

We can never prove with 100% certainty that a hypothesis is true; We can only collect evidence that supports a theory. However, testing a hypothesis can set the stage for rejecting or accepting this hypothesis within a certain confidence level.

The null hypothesis is useful because it can tell us whether the results of our study are due to random chance or the manipulation of a variable (with a certain level of confidence).

A null hypothesis is rejected if the measured data is significantly unlikely to have occurred and a null hypothesis is accepted if the observed outcome is consistent with the position held by the null hypothesis.

Rejecting the null hypothesis sets the stage for further experimentation to see if a relationship between two variables exists.

Hypothesis testing is a critical part of the scientific method as it helps decide whether the results of a research study support a particular theory about a given population. Hypothesis testing is a systematic way of backing up researchers’ predictions with statistical analysis.

It helps provide sufficient statistical evidence that either favors or rejects a certain hypothesis about the population parameter.

Purpose of a Null Hypothesis

The primary purpose of the null hypothesis is to disprove an assumption.
Whether rejected or accepted, the null hypothesis can help further progress a theory in many scientific cases.
A null hypothesis can be used to ascertain how consistent the outcomes of multiple studies are.

Do you always need both a Null Hypothesis and an Alternative Hypothesis?

The null (H0) and alternative (Ha or H1) hypotheses are two competing claims that describe the effect of the independent variable on the dependent variable. They are mutually exclusive, which means that only one of the two hypotheses can be true.

While the null hypothesis states that there is no effect in the population, an alternative hypothesis states that there is statistical significance between two variables.

The goal of hypothesis testing is to make inferences about a population based on a sample. In order to undertake hypothesis testing, you must express your research hypothesis as a null and alternative hypothesis. Both hypotheses are required to cover every possible outcome of the study.

What is the difference between a null hypothesis and an alternative hypothesis?

The alternative hypothesis is the complement to the null hypothesis. The null hypothesis states that there is no effect or no relationship between variables, while the alternative hypothesis claims that there is an effect or relationship in the population.

It is the claim that you expect or hope will be true. The null hypothesis and the alternative hypothesis are always mutually exclusive, meaning that only one can be true at a time.

What are some problems with the null hypothesis?

One major problem with the null hypothesis is that researchers typically will assume that accepting the null is a failure of the experiment. However, accepting or rejecting any hypothesis is a positive result. Even if the null is not refuted, the researchers will still learn something new.

Why can a null hypothesis not be accepted?

We can either reject or fail to reject a null hypothesis, but never accept it. If your test fails to detect an effect, this is not proof that the effect doesn’t exist. It just means that your sample did not have enough evidence to conclude that it exists.

We can’t accept a null hypothesis because a lack of evidence does not prove something that does not exist. Instead, we fail to reject it.

Failing to reject the null indicates that the sample did not provide sufficient enough evidence to conclude that an effect exists.

If the p-value is greater than the significance level, then you fail to reject the null hypothesis.

Is a null hypothesis directional or non-directional?

A hypothesis test can either contain an alternative directional hypothesis or a non-directional alternative hypothesis. A directional hypothesis is one that contains the less than (“<“) or greater than (“>”) sign.

A nondirectional hypothesis contains the not equal sign (“≠”). However, a null hypothesis is neither directional nor non-directional.

A null hypothesis is a prediction that there will be no change, relationship, or difference between two variables.

The directional hypothesis or nondirectional hypothesis would then be considered alternative hypotheses to the null hypothesis.

Gill, J. (1999). The insignificance of null hypothesis significance testing. Political research quarterly , 52 (3), 647-674.

Krueger, J. (2001). Null hypothesis significance testing: On the survival of a flawed method. American Psychologist , 56 (1), 16.

Masson, M. E. (2011). A tutorial on a practical Bayesian alternative to null-hypothesis significance testing. Behavior research methods , 43 , 679-690.

Nickerson, R. S. (2000). Null hypothesis significance testing: a review of an old and continuing controversy. Psychological methods , 5 (2), 241.

Rozeboom, W. W. (1960). The fallacy of the null-hypothesis significance test. Psychological bulletin , 57 (5), 416.

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Knowledge Base

An Easy Introduction to Statistical Significance (With Examples)

Published on January 7, 2021 by Pritha Bhandari . Revised on June 22, 2023.

If a result is statistically significant , that means it’s unlikely to be explained solely by chance or random factors. In other words, a statistically significant result has a very low chance of occurring if there were no true effect in a research study.

The p value , or probability value, tells you the statistical significance of a finding. In most studies, a p value of 0.05 or less is considered statistically significant, but this threshold can also be set higher or lower.

How do you test for statistical significance, what is a significance level, problems with relying on statistical significance, other types of significance in research, other interesting articles, frequently asked questions about statistical significance.

In quantitative research , data are analyzed through null hypothesis significance testing, or hypothesis testing. This is a formal procedure for assessing whether a relationship between variables or a difference between groups is statistically significant.

Null and alternative hypotheses

To begin, research predictions are rephrased into two main hypotheses: the null and alternative hypothesis .

A null hypothesis ( H 0 ) always predicts no true effect, no relationship between variables , or no difference between groups.
An alternative hypothesis ( H a or H 1 ) states your main prediction of a true effect, a relationship between variables, or a difference between groups.

Hypothesis testin g always starts with the assumption that the null hypothesis is true. Using this procedure, you can assess the likelihood (probability) of obtaining your results under this assumption. Based on the outcome of the test, you can reject or retain the null hypothesis.

H 0 : There is no difference in happiness between actively smiling and not smiling.
H a : Actively smiling leads to more happiness than not smiling.

Test statistics and p values

Every statistical test produces:

A test statistic that indicates how closely your data match the null hypothesis.
A corresponding p value that tells you the probability of obtaining this result if the null hypothesis is true.

The p value determines statistical significance. An extremely low p value indicates high statistical significance, while a high p value means low or no statistical significance.

Next, you perform a t test to see whether actively smiling leads to more happiness. Using the difference in average happiness between the two groups, you calculate:

a t value (the test statistic) that tells you how much the sample data differs from the null hypothesis,
a p value showing the likelihood of finding this result if the null hypothesis is true.

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The significance level , or alpha (α), is a value that the researcher sets in advance as the threshold for statistical significance. It is the maximum risk of making a false positive conclusion ( Type I error ) that you are willing to accept .

In a hypothesis test, the p value is compared to the significance level to decide whether to reject the null hypothesis.

If the p value is higher than the significance level, the null hypothesis is not refuted, and the results are not statistically significant .
If the p value is lower than the significance level, the results are interpreted as refuting the null hypothesis and reported as statistically significant .

Usually, the significance level is set to 0.05 or 5%. That means your results must have a 5% or lower chance of occurring under the null hypothesis to be considered statistically significant.

The significance level can be lowered for a more conservative test. That means an effect has to be larger to be considered statistically significant.

The significance level may also be set higher for significance testing in non-academic marketing or business contexts. This makes the study less rigorous and increases the probability of finding a statistically significant result.

As best practice, you should set a significance level before you begin your study. Otherwise, you can easily manipulate your results to match your research predictions.

It’s important to note that hypothesis testing can only show you whether or not to reject the null hypothesis in favor of the alternative hypothesis. It can never “prove” the null hypothesis, because the lack of a statistically significant effect doesn’t mean that absolutely no effect exists.

When reporting statistical significance, include relevant descriptive statistics about your data (e.g., means and standard deviations ) as well as the test statistic and p value.

There are various critiques of the concept of statistical significance and how it is used in research.

Researchers classify results as statistically significant or non-significant using a conventional threshold that lacks any theoretical or practical basis. This means that even a tiny 0.001 decrease in a p value can convert a research finding from statistically non-significant to significant with almost no real change in the effect.

On its own, statistical significance may also be misleading because it’s affected by sample size. In extremely large samples , you’re more likely to obtain statistically significant results, even if the effect is actually small or negligible in the real world. This means that small effects are often exaggerated if they meet the significance threshold, while interesting results are ignored when they fall short of meeting the threshold.

The strong emphasis on statistical significance has led to a serious publication bias and replication crisis in the social sciences and medicine over the last few decades. Results are usually only published in academic journals if they show statistically significant results—but statistically significant results often can’t be reproduced in high quality replication studies.

As a result, many scientists call for retiring statistical significance as a decision-making tool in favor of more nuanced approaches to interpreting results.

That’s why APA guidelines advise reporting not only p values but also effect sizes and confidence intervals wherever possible to show the real world implications of a research outcome.

Aside from statistical significance, clinical significance and practical significance are also important research outcomes.

Practical significance shows you whether the research outcome is important enough to be meaningful in the real world. It’s indicated by the effect size of the study.

Clinical significance is relevant for intervention and treatment studies. A treatment is considered clinically significant when it tangibly or substantially improves the lives of patients.

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Statistical significance is a term used by researchers to state that it is unlikely their observations could have occurred under the null hypothesis of a statistical test . Significance is usually denoted by a p -value , or probability value.

Statistical significance is arbitrary – it depends on the threshold, or alpha value, chosen by the researcher. The most common threshold is p < 0.05, which means that the data is likely to occur less than 5% of the time under the null hypothesis .

When the p -value falls below the chosen alpha value, then we say the result of the test is statistically significant.

A p -value , or probability value, is a number describing how likely it is that your data would have occurred under the null hypothesis of your statistical test .

P -values are usually automatically calculated by the program you use to perform your statistical test. They can also be estimated using p -value tables for the relevant test statistic .

P -values are calculated from the null distribution of the test statistic. They tell you how often a test statistic is expected to occur under the null hypothesis of the statistical test, based on where it falls in the null distribution.

If the test statistic is far from the mean of the null distribution, then the p -value will be small, showing that the test statistic is not likely to have occurred under the null hypothesis.

No. The p -value only tells you how likely the data you have observed is to have occurred under the null hypothesis .

If the p -value is below your threshold of significance (typically p < 0.05), then you can reject the null hypothesis, but this does not necessarily mean that your alternative hypothesis is true.

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What is p-value , p value vs alpha level, p values and critical values, how is p-value calculated, p-value in hypothesis testing, p-values and statistical significance, reporting p-values, our learners also ask, what is p-value in statistical hypothesis.

Few statistical estimates are as significant as the p-value. The p-value or probability value is a number, calculated from a statistical test , that describes how likely your results would have occurred if the null hypothesis were true. A P-value less than 0.5 is statistically significant, while a value higher than 0.5 indicates the null hypothesis is true; hence it is not statistically significant. So, what is P-Value exactly, and why is it so important?

In statistical hypothesis testing , P-Value or probability value can be defined as the measure of the probability that a real-valued test statistic is at least as extreme as the value actually obtained. P-value shows how likely it is that your set of observations could have occurred under the null hypothesis. P-Values are used in statistical hypothesis testing to determine whether to reject the null hypothesis. The smaller the p-value, the stronger the likelihood that you should reject the null hypothesis.

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P-values are expressed as decimals and can be converted into percentage. For example, a p-value of 0.0237 is 2.37%, which means there's a 2.37% chance of your results being random or having happened by chance. The smaller the P-value, the more significant your results are.

In a hypothesis test, you can compare the p value from your test with the alpha level selected while running the test. Now, let’s try to understand what is P-Value vs Alpha level.

A P-value indicates the probability of getting an effect no less than that actually observed in the sample data.

An alpha level will tell you the probability of wrongly rejecting a true null hypothesis. The level is selected by the researcher and obtained by subtracting your confidence level from 100%. For instance, if you are 95% confident in your research, the alpha level will be 5% (0.05).

When you run the hypothesis test, if you get:

A small p value (<=0.05), you should reject the null hypothesis
A large p value (>0.05), you should not reject the null hypothesis

In addition to the P-value, you can use other values given by your test to determine if your null hypothesis is true.

For example, if you run an F-test to compare two variances in Excel, you will obtain a p-value, an f-critical value, and a f-value. Compare the f-value with f-critical value. If f-critical value is lower, you should reject the null hypothesis.

P-Values are usually calculated using p-value tables or spreadsheets, or calculated automatically using statistical software like R, SPSS, etc.

Depending on the test statistic and degrees of freedom (subtracting no. of independent variables from no. of observations) of your test, you can find out from the tables how frequently you can expect the test statistic to be under the null hypothesis.

How to calculate P-value depends on which statistical test you’re using to test your hypothesis.

Every statistical test uses different assumptions and generates different statistics. Select the test method that best suits your data and matches the effect or relationship being tested.
The number of independent variables included in your test determines how big or small the test statistic should be in order to generate the same p-value.

Regardless of what statistical test you are using, the p-value will always denote the same thing – how frequently you can expect to get a test statistic as extreme or even more extreme than the one given by your test.

In the P-Value approach to hypothesis testing, a calculated probability is used to decide if there’s evidence to reject the null hypothesis, also known as the conjecture. The conjecture is the initial claim about a data population, while the alternative hypothesis ascertains if the observed population parameter differs from the population parameter value according to the conjecture.

Effectively, the significance level is declared in advance to determine how small the P-value needs to be such that the null hypothesis is rejected. The levels of significance vary from one researcher to another; so it can get difficult for readers to compare results from two different tests. That is when P-value makes things easier.

Readers could interpret the statistical significance by referring to the reported P-value of the hypothesis test. This is known as the P-value approach to hypothesis testing. Using this, readers could decide for themselves whether the p value represents a statistically significant difference.

The level of statistical significance is usually represented as a P-value between 0 and 1. The smaller the p-value, the more likely it is that you would reject the null hypothesis.

A P-Value < or = 0.05 is considered statistically significant. It denotes strong evidence against the null hypothesis, since there is below 5% probability of the null being correct. So, we reject the null hypothesis and accept the alternative hypothesis.
But if P-Value is lower than your threshold of significance, though the null hypothesis can be rejected, it does not mean that there is 95% probability of the alternative hypothesis being true.
A P-Value >0.05 is not statistically significant. It denotes strong evidence for the null hypothesis being true. Thus, we retain the null hypothesis and reject the alternative hypothesis. We cannot accept null hypothesis; we can only reject or not reject it.

A statistically significant result does not prove a research hypothesis to be correct. Instead, it provides support for or provides evidence for the hypothesis.

You should report exact P-Values upto two or three decimal places.
For P-values less than .001, report as p < .001.
Do not use 0 before the decimal point as it cannot equal1. Write p = .001, and not p = 0.001
Make sure p is always italicized and there is space on either side of the = sign.
It is impossible to get P = .000, and should be written as p < .001

An investor says that the performance of their investment portfolio is equivalent to that of the Standard & Poor’s (S&P) 500 Index. He performs a two-tailed test to determine this.

The null hypothesis here says that the portfolio’s returns are equivalent to the returns of S&P 500, while the alternative hypothesis says that the returns of the portfolio and the returns of the S&P 500 are not equivalent.

The p-value hypothesis test gives a measure of how much evidence is present to reject the null hypothesis. The smaller the p value, the higher the evidence against null hypothesis.

Therefore, if the investor gets a P value of .001, it indicates strong evidence against null hypothesis. So he confidently deduces that the portfolio’s returns and the S&P 500’s returns are not equivalent.

1. What does P-value mean?

P-Value or probability value is a number that denotes the likelihood of your data having occurred under the null hypothesis of your statistical test.

2. What does p 0.05 mean?

A P-value less than 0.05 is deemed to be statistically significant, meaning the null hypothesis should be rejected in such a case. A P-Value greater than 0.05 is not considered to be statistically significant, meaning the null hypothesis should not be rejected.

3. What is P-value and how is it calculated?

The p-value or probability value is a number, calculated from a statistical test, that tells how likely it is that your results would have occurred under the null hypothesis of the test.

P-values are usually automatically calculated using statistical software. They can also be calculated using p-value tables for the relevant statistical test. P values are calculated based on the null distribution of the test statistic. In case the test statistic is far from the mean of the null distribution, the p-value obtained is small. It indicates that the test statistic is unlikely to have occurred under the null hypothesis.

4. What is p-value in research?

P values are used in hypothesis testing to help determine whether the null hypothesis should be rejected. It plays a major role when results of research are discussed. Hypothesis testing is a statistical methodology frequently used in medical and clinical research studies.

5. Why is the p-value significant?

Statistical significance is a term that researchers use to say that it is not likely that their observations could have occurred if the null hypothesis were true. The level of statistical significance is usually represented as a P-value or probability value between 0 and 1. The smaller the p-value, the more likely it is that you would reject the null hypothesis.

6. What is null hypothesis and what is p-value?

A null hypothesis is a kind of statistical hypothesis that suggests that there is no statistical significance in a set of given observations. It says there is no relationship between your variables.

P-value or probability value is a number, calculated from a statistical test, that tells how likely it is that your results would have occurred under the null hypothesis of the test.

P-Value is used to determine the significance of observational data. Whenever researchers notice an apparent relation between two variables, a P-Value calculation helps ascertain if the observed relationship happened as a result of chance. Learn more about statistical analysis and data analytics and fast track your career with our Professional Certificate Program In Data Analytics .

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P-Value: Comprehensive Guide to Understand, Apply, and Interpret

A p-value is a statistical metric used to assess a hypothesis by comparing it with observed data.

This article delves into the concept of p-value, its calculation, interpretation, and significance. It also explores the factors that influence p-value and highlights its limitations.

Table of Content

What is P-value?

How P-value is calculated?

How to interpret p-value, p-value in hypothesis testing, implementing p-value in python, applications of p-value, what is the p-value.

The p-value, or probability value, is a statistical measure used in hypothesis testing to assess the strength of evidence against a null hypothesis. It represents the probability of obtaining results as extreme as, or more extreme than, the observed results under the assumption that the null hypothesis is true.

In simpler words, it is used to reject or support the null hypothesis during hypothesis testing. In data science, it gives valuable insights on the statistical significance of an independent variable in predicting the dependent variable.

Calculating the p-value typically involves the following steps:

Formulate the Null Hypothesis (H0) : Clearly state the null hypothesis, which typically states that there is no significant relationship or effect between the variables.
Choose an Alternative Hypothesis (H1) : Define the alternative hypothesis, which proposes the existence of a significant relationship or effect between the variables.
Determine the Test Statistic : Calculate the test statistic, which is a measure of the discrepancy between the observed data and the expected values under the null hypothesis. The choice of test statistic depends on the type of data and the specific research question.
Identify the Distribution of the Test Statistic : Determine the appropriate sampling distribution for the test statistic under the null hypothesis. This distribution represents the expected values of the test statistic if the null hypothesis is true.
Calculate the Critical-value : Based on the observed test statistic and the sampling distribution, find the probability of obtaining the observed test statistic or a more extreme one, assuming the null hypothesis is true.
Interpret the results: Compare the critical-value with t-statistic. If the t-statistic is larger than the critical value, it provides evidence to reject the null hypothesis, and vice-versa.

Its interpretation depends on the specific test and the context of the analysis. Several popular methods for calculating test statistics that are utilized in p-value calculations.

Test	Scenario	Interpretation
	Used when dealing with large sample sizes or when the population standard deviation is known.	A small p-value (smaller than 0.05) indicates strong evidence against the null hypothesis, leading to its rejection.
	Appropriate for small sample sizes or when the population standard deviation is unknown.	Similar to the Z-test
	Used for tests of independence or goodness-of-fit.	A small p-value indicates that there is a significant association between the categorical variables, leading to the rejection of the null hypothesis.
	Commonly used in Analysis of Variance (ANOVA) to compare variances between groups.	A small p-value suggests that at least one group mean is different from the others, leading to the rejection of the null hypothesis.
	Measures the strength and direction of a linear relationship between two continuous variables.	A small p-value indicates that there is a significant linear relationship between the variables, leading to rejection of the null hypothesis that there is no correlation.

In general, a small p-value indicates that the observed data is unlikely to have occurred by random chance alone, which leads to the rejection of the null hypothesis. However, it’s crucial to choose the appropriate test based on the nature of the data and the research question, as well as to interpret the p-value in the context of the specific test being used.

The table given below shows the importance of p-value and shows the various kinds of errors that occur during hypothesis testing.


	Correct decision based on the given p-value	Type I error
	Type II error	Incorrect decision based on the given p-value

Type I error: Incorrect rejection of the null hypothesis. It is denoted by α (significance level). Type II error: Incorrect acceptance of the null hypothesis. It is denoted by β (power level)

Let’s consider an example to illustrate the process of calculating a p-value for Two Sample T-Test:

A researcher wants to investigate whether there is a significant difference in mean height between males and females in a population of university students.

Suppose we have the following data:

$\overline{x_1} = 175$

Starting with interpreting the process of calculating p-value

Step 1 : Formulate the Null Hypothesis (H0):

H0: There is no significant difference in mean height between males and females.

Step 2 : Choose an Alternative Hypothesis (H1):

H1: There is a significant difference in mean height between males and females.

Step 3 : Determine the Test Statistic:

The appropriate test statistic for this scenario is the two-sample t-test, which compares the means of two independent groups.

The t-statistic is a measure of the difference between the means of two groups relative to the variability within each group. It is calculated as the difference between the sample means divided by the standard error of the difference. It is also known as the t-value or t-score.

$t = \frac{\overline{x_1} - \overline{x_2}}{ \sqrt{\frac{(s_1)^2}{n_1} + \frac{(s_2)^2}{n_2}}}$

s1 = First sample’s standard deviation
s2 = Second sample’s standard deviation
n1 = First sample’s sample size
n2 = Second sample’s sample size

$\begin{aligned}t &= \frac{175 - 168}{\sqrt{\frac{5^2}{30} + \frac{6^2}{35}}}\\&= \frac{7}{\sqrt{0.8333 + 1.0286}}\\&= \frac{7}{\sqrt{1.8619}}\\& \approx \frac{7}{1.364}\\& \approx 5.13\end{aligned}$

So, the calculated two-sample t-test statistic (t) is approximately 5.13.

Step 4 : Identify the Distribution of the Test Statistic:

The t-distribution is used for the two-sample t-test . The degrees of freedom for the t-distribution are determined by the sample sizes of the two groups.

The t-distribution is a probability distribution with tails that are thicker than those of the normal distribution.

where, n1 is total number of values for 1st category.
n2 is total number of values for 2nd category.

The degrees of freedom (63) represent the variability available in the data to estimate the population parameters. In the context of the two-sample t-test, higher degrees of freedom provide a more precise estimate of the population variance, influencing the shape and characteristics of the t-distribution.

T-Statistic

The t-distribution is symmetric and bell-shaped, similar to the normal distribution. As the degrees of freedom increase, the t-distribution approaches the shape of the standard normal distribution. Practically, it affects the critical values used to determine statistical significance and confidence intervals.

Step 5 : Calculate Critical Value.

To find the critical t-value with a t-statistic of 5.13 and 63 degrees of freedom, we can either consult a t-table or use statistical software.

We can use scipy.stats module in Python to find the critical t-value using below code.

Comparing with T-Statistic:

The larger t-statistic suggests that the observed difference between the sample means is unlikely to have occurred by random chance alone. Therefore, we reject the null hypothesis.

$(\alpha)$

p ≤ (α = 0.05) : Reject the null hypothesis. There is sufficient evidence to conclude that the observed effect or relationship is statistically significant, meaning it is unlikely to have occurred by chance alone.
p > (α = 0.05) : reject alternate hypothesis (or accept null hypothesis). The observed effect or relationship does not provide enough evidence to reject the null hypothesis. This does not necessarily mean there is no effect; it simply means the sample data does not provide strong enough evidence to rule out the possibility that the effect is due to chance.

In case the significance level is not specified, consider the below general inferences while interpreting your results.

If p > .10: not significant
If p ≤ .10: slightly significant
If p ≤ .05: significant
If p ≤ .001: highly significant

Graphically, the p-value is located at the tails of any confidence interval. [As shown in fig 1]

Fig 1: Graphical Representation

What influences p-value?

The p-value in hypothesis testing is influenced by several factors:

Sample Size : Larger sample sizes tend to yield smaller p-values, increasing the likelihood of detecting significant effects.
Effect Size: A larger effect size results in smaller p-values, making it easier to detect a significant relationship.
Variability in the Data : Greater variability often leads to larger p-values, making it harder to identify significant effects.
Significance Level : A lower chosen significance level increases the threshold for considering p-values as significant.
Choice of Test: Different statistical tests may yield different p-values for the same data.
Assumptions of the Test : Violations of test assumptions can impact p-values.

Understanding these factors is crucial for interpreting p-values accurately and making informed decisions in hypothesis testing.

Significance of P-value

The p-value provides a quantitative measure of the strength of the evidence against the null hypothesis.
Decision-Making in Hypothesis Testing
P-value serves as a guide for interpreting the results of a statistical test. A small p-value suggests that the observed effect or relationship is statistically significant, but it does not necessarily mean that it is practically or clinically meaningful.

Limitations of P-value

The p-value is not a direct measure of the effect size, which represents the magnitude of the observed relationship or difference between variables. A small p-value does not necessarily mean that the effect size is large or practically meaningful.
Influenced by Various Factors

The p-value is a crucial concept in statistical hypothesis testing, serving as a guide for making decisions about the significance of the observed relationship or effect between variables.

Let’s consider a scenario where a tutor believes that the average exam score of their students is equal to the national average (85). The tutor collects a sample of exam scores from their students and performs a one-sample t-test to compare it to the population mean (85).

The code performs a one-sample t-test to compare the mean of a sample data set to a hypothesized population mean.
It utilizes the scipy.stats library to calculate the t-statistic and p-value. SciPy is a Python library that provides efficient numerical routines for scientific computing.
The p-value is compared to a significance level (alpha) to determine whether to reject the null hypothesis.

Since, 0.7059>0.05 , we would conclude to fail to reject the null hypothesis. This means that, based on the sample data, there isn’t enough evidence to claim a significant difference in the exam scores of the tutor’s students compared to the national average. The tutor would accept the null hypothesis, suggesting that the average exam score of their students is statistically consistent with the national average.

During Forward and Backward propagation: When fitting a model (say a Multiple Linear Regression model), we use the p-value in order to find the most significant variables that contribute significantly in predicting the output.
Effects of various drug medicines: It is highly used in the field of medical research in determining whether the constituents of any drug will have the desired effect on humans or not. P-value is a very strong statistical tool used in hypothesis testing. It provides a plethora of valuable information while making an important decision like making a business intelligence inference or determining whether a drug should be used on humans or not, etc. For any doubt/query, comment below.

The p-value is a crucial concept in statistical hypothesis testing, providing a quantitative measure of the strength of evidence against the null hypothesis. It guides decision-making by comparing the p-value to a chosen significance level, typically 0.05. A small p-value indicates strong evidence against the null hypothesis, suggesting a statistically significant relationship or effect. However, the p-value is influenced by various factors and should be interpreted alongside other considerations, such as effect size and context.

Frequently Based Questions (FAQs)

Why is p-value greater than 1.

A p-value is a probability, and probabilities must be between 0 and 1. Therefore, a p-value greater than 1 is not possible.

What does P 0.01 mean?

It means that the observed test statistic is unlikely to occur by chance if the null hypothesis is true. It represents a 1% chance of observing the test statistic or a more extreme one under the null hypothesis.

Is 0.9 a good p-value?

A good p-value is typically less than or equal to 0.05, indicating that the null hypothesis is likely false and the observed relationship or effect is statistically significant.

What is p-value in a model?

It is a measure of the statistical significance of a parameter in the model. It represents the probability of obtaining the observed value of the parameter or a more extreme one, assuming the null hypothesis is true.

Why is p-value so low?

A low p-value means that the observed test statistic is unlikely to occur by chance if the null hypothesis is true. It suggests that the observed relationship or effect is statistically significant and not due to random sampling variation.

How Can You Use P-value to Compare Two Different Results of a Hypothesis Test?

Compare p-values: Lower p-value indicates stronger evidence against null hypothesis, favoring results with smaller p-values in hypothesis testing.

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What Does “Statistical Significance” Mean?

Statistical significance is the probability of finding a given deviation from the null hypothesis -or a more extreme one- in a sample. Statistical significance is often referred to as the p-value (short for “probability value”) or simply p in research papers. A small p-value basically means that your data are unlikely under some null hypothesis. A somewhat arbitrary convention is to reject the null hypothesis if p < 0.05 .

Example 1 - 10 Coin Flips

I've a coin and my null hypothesis is that it's balanced - which means it has a 0.5 chance of landing heads up. I flip my coin 10 times, which may result in 0 through 10 heads landing up. The probabilities for these outcomes -assuming my coin is really balanced- are shown below. Technically, this is a binomial distribution . The formula for computing these probabilities is based on mathematics and the (very general) assumption of independent and identically distributed variables . Keep in mind that probabilities are relative frequencies. So the 0.24 probability of finding 5 heads means that if I'd draw a 1,000 samples of 10 coin flips, some 24% of those samples should result in 5 heads up.

Statistical Significance in Binomial Distribution

Now, 9 of my 10 coin flips actually land heads up. The previous figure says that the probability of finding 9 or more heads in a sample of 10 coin flips, p = 0.01 . If my coin is really balanced, the probability is only 1 in 100 of finding what I just found. So, based on my sample of N = 10 coin flips, I reject the null hypothesis : I no longer believe that my coin was balanced after all. Now don't overlook the basic reasoning here: what I want to know is the chance of my coin landing heads up. This parameter is a single numberI estimate this chance by computing the proportion This chance a property of my coin: a fixed number that doesn't fluctuate in any way. What does flucI can estimate this chance by I estimated this chance This is a parameter -a single number that says something about my population of coin flips. I assumed that I drew a sample of 10 coin flips --> Statistical significance reall -->

Example 2 - T-Test

A sample of 360 people took a grammar test. We'd like to know if male respondents score differently than female respondents. Our null hypothesis is that on average, male respondents score the same number of points as female respondents. The table below summarizes the means and standard deviations for this sample.

Note that females scored 3.5 points higher than males in this sample. However, samples typically differ somewhat from populations. The question is: if the mean scores for all males and all females are equal, then what's the probability of finding this mean difference or a more extreme one in a sample of N = 360? This question is answered by running an independent samples t-test .

Test Statistic - T

So what sample mean differences can we reasonably expect ? Well, this depends on

the standard deviations and
the sample sizes we have.

We therefore standardize our mean difference of 3.5 points, resulting in t = -2.2 So this t-value -our test statistic- is simply the sample mean difference corrected for sample sizes and standard deviations. Interestingly, we know the sampling distribution -and hence the probability- for t.

1-Tailed Statistical Significance

1-tailed statistical significance is the probability of finding a given deviation from the null hypothesis -or a larger one- in a sample. In our example, p (1-tailed) ≈ 0.014 . The probability of finding t ≤ -2.2 -corresponding to our mean difference of 3.5 points- is 1.4%. If the population means are really equal and we'd draw 1,000 samples, we'd expect only 14 samples to come up with a mean difference of 3.5 points or larger. In short, this sample outcome is very unlikely if the population mean difference is zero. We therefore reject the null hypothesis. Conclusion: men and women probably don't score equally on our test. Some scientists will report precisely these results. However, a flaw here is that our reasoning suggests that we'd retain our null hypothesis if t is large rather than small. A large t-value ends up in the right tail of our distribution . However, our p-value only takes into account the left tail in which our (small) t-value of -2.2 ended up. If we take into account both possibilities, we should report p = 0.028, the 2-tailed significance.

2-Tailed Statistical Significance

2-tailed statistical significance is the probability of finding a given absolute deviation from the null hypothesis -or a larger one- in a sample. For a t test, very small as well as very large t-values are unlikely under H 0 . Therefore, we shouldn't ignore the right tail of the distribution like we do when reporting a 1-tailed p-value. It suggests that we wouldn't reject the null hypothesis if t had been 2.2 instead of -2.2. However, both t-values are equally unlikely under H 0 . A convention is to compute p for t = -2.2 and the opposite effect : t = 2.2. Adding them results in our 2-tailed p-value: p (2-tailed) = 0.028 in our example. Because the distribution is symmetrical around 0, these 2 p-values are equal. So we may just as well double our 1-tailed p-value.

1-Tailed or 2-Tailed Significance?

So should you report the 1-tailed or 2-tailed significance? First off, many statistical tests -such as ANOVA and chi-square tests - only result in a 1-tailed p-value so that's what you'll report. However, the question does apply to t-tests , z-tests and some others. There's no full consensus among data analysts which approach is better. I personally always report 2-tailed p-values whenever available. A major reason is that when some test only yields a 1-tailed p-value, this often includes effects in different directions. “What on earth is he tryi...?” That needs some explanation, right?

T-Test or ANOVA?

We compared young to middle aged people on a grammar test using a t-test . Let's say young people did better. This resulted in a 1-tailed significance of 0.096. This p-value does not include the opposite effect of the same magnitude: middle aged people doing better by the same number of points. The figure below illustrates these scenarios.

1 Tailed Statistical Significance in T-Test

We then compared young, middle aged and old people using ANOVA . Young people performed best, old people performed worst and middle aged people are exactly in between. This resulted in a 1-tailed significance of 0.035. Now this p-value does include the opposite effect of the same magnitude.

1 Tailed Statistical Significance in ANOVA F-Test

Now, if p for ANOVA always includes effects in different directions, then why would you not include these when reporting a t-test? In fact, the independent samples t-test is technically a special case of ANOVA: if you run ANOVA on 2 groups, the resulting p-value will be identical to the 2-tailed significance from a t-test on the same data. The same principle applies to the z-test versus the chi-square test.

The “Alternative Hypothesis”

Reporting 1-tailed significance is sometimes defended by claiming that the researcher is expecting an effect in a given direction. However, I cannot verify that. Perhaps such “alternative hypotheses” were only made up in order to render results more statistically significant. Second, expectations don't rule out possibilities . If somebody is absolutely sure that some effect will have some direction, then why use a statistical test in the first place?

Statistical Versus Practical Significance

So what does “statistical significance” really tell us? Well, it basically says that some effect is very probably not zero in some population. So is that what we really want to know? That a mean difference, correlation or other effect is “not zero”? No. Of course not. We really want to know how large some mean difference, correlation or other effect is. However, that's not what statistical significance tells us. For example, a correlation of 0.1 in a sample of N = 1,000 has p ≈ 0.0015. This is highly statistically significant : the population correlation is very probably not 0.000... However, a 0.1 correlation is not distinguishable from 0 in a scatterplot . So it's probably not practically significant . Reversely, a 0.5 correlation with N = 10 has p ≈ 0.14 and hence is not statistically significant. Nevertheless, a scatterplot shows a strong relation between our variables. However, since our sample size is very small, this strong relation may very well be limited to our small sample: it has a 14% chance of occurring if our population correlation is really zero.

The basic problem here is that any effect is statistically significant if the sample size is large enough. And therefore, results must have both statistical and practical significance in order to carry any importance. effect size . --> Confidence intervals nicely combine these two pieces of information and can thus be argued to be more useful than just statistical significance.

Thanks for reading!

The basic problem here is that any effect is statistically significant if the sample size is large enough. And therefore, we should take into account both statistical and practical significance when evaluating results. We should therefore only

Tell us what you think!

This tutorial has 14 comments:.

By KEFALE on May 26th, 2021

God may bless you

By John Xie on January 12th, 2022

A p-value, or a confidence interval maybe referred to for claiming the so-called "statistical significance". However, both are continuous variables. Any attempt to dichotomize or categorize a continuous variable will be logically not defensible. Therefore, forget about "Statistically significance" entirely!

By John Ametefe on October 15th, 2022

By Lazarus Nweke on April 25th, 2023

It is very obvious that Test of significance is a bedrock of research works.

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Published: 03 September 2024

Stage dependence of Elton’s biotic resistance hypothesis of biological invasions

Kun Guo ORCID: orcid.org/0000-0001-9597-2977 1 ,
Petr Pyšek ORCID: orcid.org/0000-0001-8500-442X 2 , 3 ,
Milan Chytrý ORCID: orcid.org/0000-0002-8122-3075 4 ,
Jan Divíšek ORCID: orcid.org/0000-0002-5127-5130 4 , 5 ,
Martina Sychrová 4 , 5 ,
Zdeňka Lososová 4 ,
Mark van Kleunen ORCID: orcid.org/0000-0002-2861-3701 6 , 7 ,
Simon Pierce 8 &
Wen-Yong Guo ORCID: orcid.org/0000-0002-4737-2042 1 , 9 , 10

Nature Plants ( 2024 ) Cite this article

Metrics details

Biodiversity
Invasive species

Elton’s biotic resistance hypothesis posits that species-rich communities are more resistant to invasion. However, it remains unknown how species, phylogenetic and functional richness, along with environmental and human-impact factors, collectively affect plant invasion as alien species progress along the introduction–naturalization–invasion continuum. Using data from 12,056 local plant communities of the Czech Republic, this study reveals varying effects of these factors on the presence and richness of alien species at different invasion stages, highlighting the complexity of the invasion process. Specifically, we demonstrate that although species richness and functional richness of resident communities had mostly negative effects on alien species presence and richness, the strength and sometimes also direction of these effects varied along the continuum. Our study not only underscores that evidence for or against Elton’s biotic resistance hypothesis may be stage-dependent but also suggests that other invasion hypotheses should be carefully revisited given their potential stage-dependent nature.

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Data availability

The data used in this study were obtained from these sources: the data on vegetation plots were from the Czech National Phytosociological Database 54 ( https://botzool.cz/vegsci/phytosociologicalDb/ ); species’ statuses along the invasion continuum were extracted from Pyšek et al. 55 ; the three leaf traits required for CSR calculation were collected from the Pladias Database of the Czech Flora and Vegetation 58 and other publications 61 , 62 , 63 , 64 , 65 , 66 ; species CSR scores were calculated using the StrateFy tool 60 ; climatic variables were extracted from Tolasz 73 ; soil pH was collected from the Land Use/Land Cover Area Frame Survey 74 ; and the human population density of the cadastral area where each plot located was obtained from the Digital Vector Database of Czech Republic ArcČR v.4.0 (ref. 75 ). The data that support the findings of this study are available via GitHub at https://github.com/kun-ecology/BioticResistance_InvasionContinuum and via Zenodo at https://doi.org/10.5281/zenodo.12818669 (ref. 79 ).

Code availability

R functions for the computation of phylogenetic and functional metrics have been deposited on GitHub ( https://github.com/kun-ecology/ecoloop ). R scripts for reproducing the analyses and figures are available via GitHub at https://github.com/kun-ecology/BioticResistance_InvasionContinuum and via Zenodo at https://doi.org/10.5281/zenodo.12818669 (ref. 79 ).

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Acknowledgements

K.G. and W.-Y.G. were supported by the Natural Science Foundation of China (grant no. 32171588, awarded to W.-Y.G.) and the Shanghai Pujiang Program (grant no. 21PJ1402700, awarded to W.-Y.G.). K.G. was also supported by the Shanghai Sailing Program (grant no. 22YF1411700) and the Natural Science Foundation of China (grant no. 32301386). P.P. was supported by the Czech Science Foundation (EXPRO grant no. 19-28807X) and the Czech Academy of Sciences (long-term research development project RVO 67985939). M.C. and Z.L. were supported by the Czech Science Foundation (EXPRO grant no. 19-28491X). J.D. was supported by the Technology Agency of the Czech Republic (grant no. SS02030018). M.S. was funded by the project GEOSANT with the funding organization Masaryk University (MUNI/A/1469/2023).

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Kun Guo & Wen-Yong Guo

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Guo, K., Pyšek, P., Chytrý, M. et al. Stage dependence of Elton’s biotic resistance hypothesis of biological invasions. Nat. Plants (2024). https://doi.org/10.1038/s41477-024-01790-0

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Barriers to digital services trade and export efficiency of digital services.

1. Introduction

2. literature review, 3. research design, 3.1. model construction, 3.2. data description, 4. empirical analysis, 4.1. stochastic frontier gravity model, 4.2. trade inefficiency model, 4.3. robustness test, 5. extended analysis, 6. discussion and conclusions, 6.1. discussion, 6.2. conclusions, 6.3. recommendations, author contributions, institutional review board statement, informed consent statement, data availability statement, conflicts of interest.

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Click here to enlarge figure

Variable	Time-Invariant Model		Time-Variant Model
Variable	Coefficient	t-Value	Coefficient	t-Value
	0.760 ***	34.90	0.658 ***	28.62
	0.291 ***	8.31	0.193 ***	5.48
	0.848 ***	39.63	0.771 ***	34.49
	0.174 ***	5.09	0.0921 **	2.63
	−1.179 ***	−34.62	−1.150 ***	−33.08
	1.140 ***	8.09	1.231 ***	8.36
Constant	−7.836 ***	−12.15	−1.367	−0.98
μ	4.267 ***	13.39	6.626 ***	5.28
γ	0.885 ***	197.10	0.897 ***	215.57
η			0.006 ***	5.48
N	12,532		12,532
LR1	538.289		p-value	0.000
LR2	770.481		p-value	0.000
LR3	68.996		p-value	0.000

Stochastic Frontier Gravity Model			Trade Inefficiency Model
Variable	Coefficient	t-Value	Variable	Coefficient	t-Value
	0.780 ***	100.24		−0.186	−1.09
	0.801 ***	44.60		0.980 ***	3.28
	0.860 ***	117.83		−1.765 ***	−5.15
	0.560 ***	38.28		2.904 ***	6.72
	−1.072 ***	−92.51		−4.117 ***	−5.42
	0.745 ***	15.89	Constant	0.805 *	2.13
Constant	−21.69 ***	−74.67	N	12,532
LR	354.150		p-value	0.000

Variable	(1)		(2)		(3)		(4)
Variable	Coefficient	t-Value	Coefficient	t-Value	Coefficient	t-Value	Coefficient	t-Value
	−0.175	−1.15	−0.066	−0.30	−0.074	−0.59	−0.153	−1.10
	0.806 ***	2.87	1.213 **	2.51	0.649 ***	3.86	0.612 ***	2.74
	−1.396 ***	−4.99	−1.477 ***	−4.08	−2.787 ***	−7.40	−1.468 ***	−5.44
	2.506 ***	6.44	2.852 ***	4.97	1.472 ***	11.52	2.390 ***	7.38
	−3.282 ***	−4.93	−3.567 ***	−3.81	−3.992 ***	−11.29	−3.244 ***	−5.57
Constant	1.144 ***	3.63	1.282 ***	3.73	1.829 ***	10.12	1.153 ***	4.00
N	9,193		3,982		5,211		11,473

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Wang, X.; Zhang, J.; Zhu, Y. Barriers to Digital Services Trade and Export Efficiency of Digital Services. Sustainability 2024 , 16 , 7517. https://doi.org/10.3390/su16177517

Wang X, Zhang J, Zhu Y. Barriers to Digital Services Trade and Export Efficiency of Digital Services. Sustainability . 2024; 16(17):7517. https://doi.org/10.3390/su16177517

Wang, Xiaomei, Jia Zhang, and Yixin Zhu. 2024. "Barriers to Digital Services Trade and Export Efficiency of Digital Services" Sustainability 16, no. 17: 7517. https://doi.org/10.3390/su16177517

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